How do we identify twin primes .

There is a trivial algorithm. All twin primes produce composites of the form $X^2-1$. An interesting property of even perfect squares minus 1 (which are always composite) is the triviality of their smallest prime factor unless they are twin-prime composites. This makes it extremely fast to factor them and easy to determine the instances of twin primes (simply by elimination). The rule is that the smallest prime factor of a non-twin-prime $X^2-1$ composite cannot be greater than the square root of its square root - and usually much smaller. If such a factor is not found, the composite must be the product of twin primes. Thus the largest of these non-twin-prime factors less than $10^{12}$ is $991$ for $999836006723$.

I wrote an Excel program that exploits this. http://www.naturalnumbers.org/TwinPrimeCalc.xlsm